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5902S. E. Harper et al. resulting in a relative calibration better than 1 per cent in total. The calibration to the astronomical brightness scale is done using daily observations of Tau A and Cas A using the WMAP derived models for the flux density of these sources (Weiland et al. 2011). The bandpass-weighted central frequency of the C-BASS North intensity data is 4.76 GHz when calibrated to a flat-spectrum (β = −2 ) source. Over the realistic range of source spectral indices observed in the C-BASS map (−3.5 < β < −2), colour corrections will contribute less than a 1 per cent uncertainty. One remaining source of uncertainty is the primary beam deconvolution; tests of sources flux densities in the final C-BASS map suggest that the deconvolution (Section 2.3.2) has an additional 1 per cent uncertainty. However, at present, the C-BASS calibration is still being finalized in preparation for the public release of the survey (Taylor et al. in preparation), as such we adopt a conservative 3 per cent calibration uncertainty for this work.

2.3 C-BASS maps

2.3.1 Map making

To produce the C-BASS maps, we use the destriping map-maker Descart (Sutton et al. 2010). We use a destriping offset length of 5 s to remove any large-scale 1/f noise in the data. The C-BASS map does not include day-time data due to the impact of the Sun in the far-sidelobes of the beam on the large-scale structures in the map. The final map has a 0°.73 FWHM, with a sensitivity of approximately 0.25 mK/beam (instrumental white noise only). To estimate the level of the residual 1/f noise in the map, we performed a jack-knife test where we split the C-BASS observations into two approximately equal-sized data sets, and produced a map for each. Differencing the two maps, we compared the ratio of the root mean squared (RMS) estimated in the residual data (which contains both residual 1/f and white noise) to the expected RMS assuming just white noise. We find that on average the C-BASS map has a 10 per cent excess of residual 1/f noise at scales of a few degrees and larger, which is negligible relative to other sources of uncertainty.

2.3.2 Deconvolution

The C-BASS beam is diffraction-limited with a main beam efficiency of 72.8 per cent (i.e. the power within the first null, which is at 1°.0). The sidelobe structures of the C-BASS beam are imprinted into the final map and result in an effective calibration which varies with angular scale. In previous C-BASS papers (Irfan et al. 2015; Dickinson et al. 2019; Cepeda-Arroita et al. 2021), this resulted in an effective calibration uncertainty due to the beam of approximately 5 per cent.

We have accurately characterized the C-BASS beam using detailed physical optics simulations verified by observations of bright point sources and direct beam measurements using a radio transmitter, improving the measurements presented by Holler et al. (2013). This allows us to deconvolve the effect of the beam, resulting in an effective Gaussian beam and a window function that is largely flat in log-harmonic space.

The deconvolution of the C-BASS map is done in spherical harmonic space using the routines provided by HEALPix (Górski et al. 2005). First, we transform the C-BASS beam model into a spherical harmonic transfer function (Bℓ). Next, we generate the transfer function for the 1°.0 Gaussian beam that we wish to smooth to (Gℓ) and divide this by the derived C-BASS beam transfer function. More details of the beam transfer function can be found in the CBASS northern survey paper (Taylor et al. in preparation). Finally, we multiply the ratio of the transfer function with the spherical harmonic amplitudes of the map such that the deconvolved C-BASS map is

$$m(\theta,\phi) = \sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell} a_{\ell,m} \frac{G_\ell}{B_\ell} Y_\ell^m(\theta,\phi),$$ (1)

where m(θ, φ) is the map along the line of sight defined by θ and φ, Gℓ is a 1°. 0 Gaussian beam transfer function, Bℓ is the C-BASS beam transfer function, and Yℓm (θ, φ) are spherical harmonics. We find that after beam deconvolution the calibration of the map on all angular scales is 1 per cent or better, with the remaining uncertainty due to small asymmetries in the beam.

2.3.3 Background source subtraction

After deconvolution of the map, we subtract a model of the extragalactic background point sources. At the frequency and resolution of the C-BASS data the sky is confusion-limited at the level of ≈0.7 mK deg−1 (Section 4.5 for details), which is several times greater than the instrumental noise level in the map. To remove the sources, we use a combination of catalogues. We use the C-BASSderived point source catalogue described in Grumitt et al. (2020) for all sources that are detected at 10 σ or better and do not lie within |b| < 1◦ of the Galactic plane. For fainter sources, or sources with a poor detection using C-BASS itself, we use several C-band point source catalogues: the Green Bank 6 cm (GB6) survey (Gregory et al. 1996), the Parkes-MIT-NRAO (PMN) survey (Wright et al. 1994) (for low declinations), and the RATAN-800 survey (Mingaliev et al. 2007) for sources missed by the GB6 survey around the North Celestial Pole (NCP).

The sources’ flux densities were binned in aℓm space and multiplied by the beam transfer function as

$$a_{\ell m} = \sum_{i=1}^{N_s} S_i Y_\ell^m(\theta_i,\phi_i) B_\ell,$$ (2)

where Si is the source flux density from one of the catalogues, Ns is the number of sources in the catalogue, Bℓ is the transfer function for the deconvolved C-BASS map at 1° resolution, and Yℓm are spherical harmonics. We note that for some sources, specifically those near the Galactic plane or with large flux densities, this method will not result in the perfect subtraction of the source because of small pointing offsets, changes in source flux densities, or systematic errors in the measurement of the source flux density. In this analysis, we mask bright sources with S4.76GHz > 1 Jy. Section 4.3 provides more details on masking.

3 ANCILLARY DATA

Here, we provide an overview of all the ancillary data sets. All data sets are smoothed to a common resolution of 1° FWHM, using beam transfer functions where available, i.e. for C-BASS, WMAP, and Planck. For WMAP and Planck maps, the CMB has been subtracted using the Spectral Matching Independent Component Analysis (SMICA) estimate from Planck Collaboration XII (2014). A summary of all the data sets used is given in Table 1. MNRAS 513, 5900–5919 (2022)

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